sigma_one_six
plain-language theorem explainer
The sum-of-divisors function sigma_1 applied to 6 equals 12, matching the explicit sum of its divisors. Number theorists building arithmetic-function layers in Recognition Science would cite this for concrete verification of the sigma definition. The proof is a one-line wrapper that applies native_decide to evaluate the expression directly.
Claim. $σ_1(6) = 12$, where $σ_k(n)$ denotes the sum of the $k$-th powers of all positive divisors of $n$.
background
The module supplies lightweight wrappers around Mathlib arithmetic functions, with sigma defined as the sum-of-divisors function σ_k via the abbrev sigma (k : ℕ) : ArithmeticFunction ℕ := ArithmeticFunction.sigma k. A separate sigma appears in the AbileneParadox module as an agent-level charge measuring the gap between private preference and public vote, but that definition is not invoked here. The local theoretical setting keeps statements minimal to stabilize basic interfaces before layering Dirichlet algebra or inversion formulas.
proof idea
The proof is a one-line wrapper that applies native_decide to compute sigma 1 6 and confirm equality with 12.
why it matters
This supplies a basic verified instance of the sum-of-divisors function inside the arithmetic-functions module. It supports incremental construction of number-theoretic tools that may later connect to Recognition Science structures such as the phi-ladder or Recognition Composition Law, though the declaration currently has no downstream uses. It functions as a low-level consistency check rather than a chain step in the T0-T8 forcing sequence.
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